Is mathematics empirical?

[Aquamarine]

There is no absolute truth in mathematics and logics. There are many consistent mathematical systems, consistent in the sense that they are not breaking the basic rules. But there is no absolute way of knowing that the basic rules are true. And there are limitless basic rules that can be chosen. Different basic rules gives different systems, like the intuitionist logic,predicate logic or fuzzy logic.

So why then is mathematics/logic interesting and why are some systems studied instead of others? I would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.

[matt grime]

I wouldn't use the word true, and I would never use a comparitive form for it in this sense. It [the set of rules/things we use] is not "more true", but it is more reasonable, and better yet more *useful* than anything else we have. That is why we use the things we do in the models we make: it works. If there were a better one, we'd use that.

[Aquamarine]

I wouldn't use the word true, and I would never use a comparitive form for it in this sense. It [the set of rules/things we use] is not "more true", but it is more reasonable, and better yet more *useful* than anything else we have. That is why we use the things we do in the models we make: it works. If there were a better one, we'd use that.
I tend to agree, but then I see little difference between "truth" and "useful".

Since there is no known ultimate cause and therefore "truth", I think the best we can get is more "useful". In the sense that it allows us to better manipulate the world.

Regarding the current prefered system, predicate logic. Suppose physicists start making better predictions using some theories based on intuitionist logic. Would not this require that most of current mathematical proofs be examined again, and accepted or discarded depending on how they fare in this new logical system that seems better suited to the real world?

[CrankFan]

So truth in mathematics is ultimately derived from physics.

How do you derive from physics, the truth that \pi is transcendental?

Regarding the current prefered system, predicate logic. Suppose physicists start making better predictions using some theories based on intuitionist logic.

Can you provide an example of what you have in mind? I seem to recall that intuitionistic theories are weaker than classical theories, any theorem of the former is a theorem of the later, but I may be wrong. An example of what you have in mind might make things clearer.

[Aquamarine]

How do you derive from physics, the truth that \pi is transcendental?
Pi is transcendental according to the basic rules chosen. Physics (and in extension the real world) do not affect the theorems that can be proven when the basic rules have been chosen. The real world affects which basic rules to chose.

Can you provide an example of what you have in mind? I seem to recall that intuitionistic theories are weaker than classical theories, any theorem of the former is a theorem of the later, but I may be wrong. An example of what you have in mind might make things clearer.
We can look at an example higher up in hierarchy. Euclidean geometry was long considered the only possible system and that it also reflected the real world, restricting mathematical research to within this area. Then is was discovered that non-euclidean geometry could be constructed in logically consistent way. And also that physicists can make better predictions assuming non-Euclidean geometry. Since then, much research has been done in non-Euclidean geometry, probably much more than in Euclidean geometry.

Regarding intuitionist logic, let us assume that it is found that the most basic fundamental particles seems to follow the basic rules of intuitionist logic better than predicate logic. That when constructing theories and when using
Reductio_ad_absurdum (http://en.wikipedia.org/wiki/Reductio_ad_absurdum) one gets poor predictions. But if instead if using intuitionist logic one can construct simple theories that give good predictions. I think this would lead to a paradigm shift in mathematics, away from predicate to intuitionist logic.

[matt grime]

OK, next challenge - can we have one physical theory constructed using reductio ad absurdum?

Surely, since physics is attempting to explain what is there, as opposed to what one may show may exist, irrespective of its apparent use, or lack of, physics is reasonably intuitionist.

And, as someone has pointed out, anything that can be proved true in intuitionistic logic is true in predicate logic.

Fuzzy logic already is used in some of the applied science, as are other logic systems.

[HallsofIvy]

Why do you choose physics specifically? If calculus is used in economics would you say that it works because of physics?? As far as your "basic rules" are concerned, the whole point of mathematics is that all statements in mathematics are of the form "if A then B". One doesn't have to know IF A is true or not, only follow the consequences IF it were true.

[Aquamarine]

Why do you choose physics specifically? If calculus is used in economics would you say that it works because of physics??
I say physics since this since some people here question whether economics or psychology is science and I do not want to discuss that in this thread. However, with physics I mean empirical science. My point is that mathematics is also an empirical science, but one step further removed from the real world than for example physics. Another step away would be logic. And logic is not automatically true, many different kinds of logics can be constructed. For example:
http://en.wikipedia.org/wiki/Sequent_calculus

As far as your "basic rules" are concerned, the whole point of mathematics is that all statements in mathematics are of the form "if A then B". One doesn't have to know IF A is true or not, only follow the consequences IF it were true.
Yes. But if you follow the the sequence backward you will find axioms that have been picked among many possible other axioms. There is no ultimate principle from which everything can be derived. And again, my point is that is the real world that determines which axioms to pick. Those that help makes better predictions are chosen. Chosen in the sense that mathematicians concentrate most of their efforts on those mathematics.

OK, next challenge - can we have one physical theory constructed using reductio ad absurdum?

Surely, since physics is attempting to explain what is there, as opposed to what one may show may exist, irrespective of its apparent use, or lack of, physics is reasonably intuitionist.

And, as someone has pointed out, anything that can be proved true in intuitionistic logic is true in predicate logic.

Fuzzy logic already is used in some of the applied science, as are other logic systems.
12-05-2004 05:37 PM
One could maybe argue that predicate logic have already failed in physics. That wave-particle duality would be an example.

I do not know enough physics or logic to know if QM is consistent with predicate logic. But if not, another logic would seems to be better choice since QM seems to make the best current predictions in its field of explanation.

Another interesting idea. What if the it turns out that that real world is discrete, that there is no infinity or continuity in nature. Would this mean that the current concept of limit and differentiation would have to be reworked?

[matt grime]

What has wave particle duality to do with predicate logic? Apart from your opinion that anything that *ought* to be dichotomic is the same as predicate calculus. The answer by the way is nothing.

As was pointed out in another thread, discrete models of space currently don't work, and for the umpteenth time limits and derivatives are mathematics, they are not physical objects! The nature of the real world would not invalidate it, in fact it is quite obviously independent of the nature of the physical world in some sense.

QM in case you didnt' notice has not stopped Newton's Laws of Motion from still being taught and used.

[HallsofIvy]

Pi is transcendental according to the basic rules chosen. Physics (and in extension the real world) do not affect the theorems that can be proven when the basic rules have been chosen. The real world affects which basic rules to chose.

Then I think the problem is that you have only studied mathematics that was designed for applications (and you seem to be using the word "physics" in a way I find peculiar). I know many forms of mathematics in which the axioms (what you call the "basic rules") have nothing to do with "the real world", for example the various finite geometries. Of course, those are not then used in applications.

I think you are looking through the "wrong end of the telescope". Mathematics, in its truest sense, does not derive axioms from "the real world". Of course, when you want to apply mathematics to "the real world", you choose the particular types of mathematics whose axioms do, in fact, appear to correspond to the real world.
Sometimes, of course, it happens that newly discovered properties of the "real world" turn out to correspond to axioms of forms of mathematics that had been developed previously without reference to the real world. I'm thinking in particular of the application to General Relativity of Riemannian spaces which were develope before the "basic rules" of General Relativity were known.

[Aquamarine]

What has wave particle duality to do with predicate logic? Apart from your opinion that anything that *ought* to be dichotomic is the same as predicate calculus. The answer by the way is nothing.
Actually there seem to have been a quite interesting discussion about this:
http://en.wikipedia.org/wiki/Is_logic_empirical%3F
http://en.wikipedia.org/wiki/Quantum_logic

As was pointed out in another thread, discrete models of space currently don't work, and for the umpteenth time limits and derivatives are mathematics, they are not physical objects! The nature of the real world would not invalidate it, in fact it is quite obviously independent of the nature of the physical world in some sense. QM in case you didnt' notice has not stopped Newton's Laws of Motion from still being taught and used.
I am not saying that the universe is discrete or continuous, in my limited understanding that is one point where QM and GR disagree.

It is true that the current limits and derivatives would not be logically falsified if the universe proved discrete. They are consistent according to the basic rules chosen. My point was that if physicists could make better predictions with limits and derivatives defined somewhat differently, they and mathematicians would abandon the current limits and derivatives. Hypothetically, a discrete universe could simulate a continuous universe at the macro level while being discrete at the micro level. The interesting predictions would be in the borderland, where the discrete and continuous mix. If the new limits and derivatives would make better predictions here than the old which ignore the discrete, they would be better suited for the real world.

Again, I am not saying this will be. Just an example of how mathematics could change which axioms to study, similar to the shift from Euclidean to non-Euclidean before.

[Aquamarine]

Then I think the problem is that you have only studied mathematics that was designed for applications (and you seem to be using the word "physics" in a way I find peculiar). I know many forms of mathematics in which the axioms (what you call the "basic rules") have nothing to do with "the real world", for example the various finite geometries. Of course, those are not then used in applications.

I think you are looking through the "wrong end of the telescope". Mathematics, in its truest sense, does not derive axioms from "the real world". Of course, when you want to apply mathematics to "the real world", you choose the particular types of mathematics whose axioms do, in fact, appear to correspond to the real world.
Sometimes, of course, it happens that newly discovered properties of the "real world" turn out to correspond to axioms of forms of mathematics that had been developed previously without reference to the real world. I'm thinking in particular of the application to General Relativity of Riemannian spaces which were develope before the "basic rules" of General Relativity were known.
Yes, there are areas of mathematics that seem to have no connection with the real world today But most mathematics are used and studied because it is useful in the real world. If it was not useful, why should society spend resources on it? Then society could as well spend the resources on teaching dead languages. So if mathematicians want to keep their status and pay, they have to choose to study systems built from axioms suited to the real world.

Regarding the second paragraph: Sometimes theory is before experiments. Sometimes experiments are before theory. Still, in the end both experiments and theory must fit together.

[matt grime]

Well, that's an interesting opinion, but factually flawed from my experience of being employed by mathematics departments (I would estimate in my current department less than 1/2 do maths that is applied to the real world, and it's predominantly an applied maths department too; this is what engineering departments are for). There is nothing to suggest all maths must eventually translate to the physical world (or it'll have it's funding cut). Intellectual rigour need not have a practical direct use.

Sometimes theory never meets experiment, thank goodness.

Interestingly I appear to be an impossible object in your opinion - a mathematician who's paid, by the state, to do mathematics that isn't practically applicable. At least it's nice when people tell me what I "have" to be.

Good to see you're so against all the higher aspects of humanity at least - utility should be one of the last things we think about in lots of cases. As the old saying goes, it'll be a great day when the schools are fully funded by the state and the military has to hold a bring and buy say to pay for its aircraft carriers.

[selfAdjoint]

matt, I want to bring up another sense of empirical. Take a mathematician who sets out to create something new - that is what math research is. The guy has in his mind a conspectus of prior mathematics in his area and he considers the relationships he knows about. Sometimes ther is a little "mix-and-match" novelty he can do, sometimes it's "monkey see monkey do" analogy with other math, and sometimes by building out a train of thought with these humble techniques he can come to a point where insight supervenes and he makes a discovery. (I want to apologize here to all the excellent women mathematicians for my choice of pronouns).

Now isn't a creation path like this "empirical" in its relationship to the world of existing mathematics, which to a mathematician is as real and present as the sensual world?

[matt grime]

I missed the post replying to me quoting back my post.

Lots of quantum things in general (such as quantum groups, cohomology etc) use quantum in a certain mathematical sense.

I still stand by my assertion that wave particle duality and predicate logic are not necessarily linked, but that is a philosophical position about the nature of logic (and mathematics) not being physical entitites. I would take it to say that the ideas of QM have led to a new way of thinking about things by introducing measurements of failure to commute (deformations), rather than it being the other way round, but again that is a personal opinion about philosophy.

However it is all hypothetical isn't it?

Mathematicians are obtuse people, they will study what they want to. Some will make connections to the real world some won't.

The development of mathematics and physics has to varying degrees been linked throughout its course. It diverged strongly in the 40's, but they are starting to come together now, it appears.

I don't think calculus wll die (something I'm a little sad about...)

It may interest you to know that there is a non-analytic concept of derivative; we algebraists have been using it for years. It uses something called the space of Dual Numbers.


Self Adjoint: I haven't actually used the word empirical directly, or addressed what may or may not be meant by it.

[CrankFan]

I do not know enough physics or logic to know if QM is consistent with predicate logic.

What does it mean for an arbitrary physical theory to be "consistent" with an arbitrary mathematical theory?

Another interesting idea. What if the it turns out that that real world is discrete, that there is no infinity or continuity in nature. Would this mean that the current concept of limit and differentiation would have to be reworked?

No.

[Aquamarine]

Well, that's an interesting opinion, but factually flawed from my experience of being employed by mathematics departments (I would estimate in my current department less than 1/2 do maths that is applied to the real world, and it's predominantly an applied maths department too; this is what engineering departments are for). There is nothing to suggest all maths must eventually translate to the physical world (or it'll have it's funding cut). Intellectual rigour need not have a practical direct use.

Sometimes theory never meets experiment, thank goodness.

Interestingly I appear to be an impossible object in your opinion - a mathematician who's paid, by the state, to do mathematics that isn't practically applicable. At least it's nice when people tell me what I "have" to be.

Good to see you're so against all the higher aspects of humanity at least - utility should be one of the last things we think about in lots of cases. As the old saying goes, it'll be a great day when the schools are fully funded by the state and the military has to hold a bring and buy say to pay for its aircraft carriers.
:biggrin:

You may not have noticed but we live in a world of limited resources. These should be allocated where they are most useful. One must always think of the alternative use, for example saving lives in health-care with more money and intelligent people.

That many must work so that a few can spend their time in useless games seems absurd. And if theoretical mathematics is of no use in the real world, then it is no better than spending time with a complex computer game. Interesting maybe for the individual, but not something others should be forced to support. Regarding the state, is is noted for its inefficiency in all areas due to lack of competition.

But of course theoretical mathematics with no apparent utility today is much better than a computer game. Exactly because it may be of use in the future. So that is one of the better ways the state spends its money, compared to many others which often even have a negative effect. But note that since it is possible to construct and study countless useless mathematics which could consume all resources in the world, some calculation regarding future use should be made when giving money to today useless mathematics.

Regarding the use of mathematics, it is of course used today in most of society. And most of those using mathematics are certainly not using axioms with no connection to the real world.

[Aquamarine]

What does it mean for an arbitrary physical theory to be "consistent" with an arbitrary mathematical theory?

Two arbitrarily chosen theories are mostly useless without some connection to the real world, for example data from experiments.

If we have data, then we can construct and test a physical theory using a mathematical theory. Note that the mathematical theory will restrict which physical theories are possible to make or affect how good the predictions are or how complex the physical theory must be to fit the data.

It would be quite difficult to construct a good physical theory that fits the data if one were forced to work only within the mathematical theory of Euclidean geometry.

[CrankFan]

Two arbitrarily chosen theories are mostly useless without some connection to the real world, for example data from experiments.

What makes you think THE theory of predicate calculus is related to QM in some obvious way? As far as I can tell, substituting "theory of recursive functions" or "theory of formal languages" in place of "theory of predicate calculus" is an arbitrary choice.


If we have data, then we can construct and test a physical theory using a mathematical theory. Note that the mathematical theory will restrict which physical theories are possible to make or affect how good the predictions are or how complex the physical theory must be to fit the data.

We all know that mathematics can be used as a tool to build physical theories. What I'm wondering is; what exactly is the process you used to come to the determination that predicate calculus is false based on the empirical evidence that suggests that light has a dual nature?

[matt grime]

There is no absolute truth in mathematics and logics.

would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.


Can we get back to the original post? It seems that this isn't a discussion about mathematics or anything in particular of that nature. It is purely a debate about what people mean by "true". And in particular comparitive ideas.

[Aquamarine]

What makes you think THE theory of predicate calculus is related to QM in some obvious way? As far as I can tell, substituting "theory of recursive functions" or "theory of formal languages" in place of "theory of predicate calculus" is an arbitrary choice.

We all know that mathematics can be used as a tool to build physical theories. What I'm wondering is; what exactly is the process you used to come to the determination that predicate calculus is false based on the empirical evidence that suggests that light has a dual nature?
That question has been asked and answered earlier. And I did not claim that QM has falsified predicate logic.

[Aquamarine]

Can we get back to the original post? It seems that this isn't a discussion about mathematics or anything in particular of that nature. It is purely a debate about what people mean by "true". And in particular comparitive ideas.
You are misquoting my post.

[Nereid]

Reading this thread for the first time, in one sitting, it seems surreal ... an awful lot of people talking past each other ...

Aquamarine has, IMHO, posed some very interesting questions, and certainly in the right place in PF.

First though, some personal clarifications - economics is just as much a 'science' as physics is (the scientific method is applied just as rigourously). Whatever 'reality' the latest and best physics or economics theories suggest, the maths used to describe those theories has its own, independent 'reality' (and if a successful theory can be described using several different mathematical frameworks - insert your favourite examples here - isn't this a practical demonstration of what I just said?). Throughout the history of maths and science (go back as far as you like ... even pre-Greece), there has been a fertile interchange between the two; for a great many of those who have contributed to our current body of theory - directly or indirectly - it would be hard to make a clear distinction; for others, their working lives have had only the most tenuous connection with the other domain (tho a detailed study of how mathematicians have worked may show a greater influence of 'external' ideas than is popularly believed).

So, to what I see as the core of Aquamarine's post: of the truly vast potential 'space' mathematicians could explore (workers in the 'truth mines' - anyone read the SF novel "Diaspora"?), the regions most heavily explored have tended to be those with an apparent connection to 'models of the physical universe'. This is NOT to say that all (or nearly all) mathematicians work in areas that are seen to be (potentially) 'useful', nor that there aren't wonderful results far from the beaten track (e.g. Cantor's work?).

Perhaps we could use the scientific method to examine this idea? I read somewhere that there are ~250,000 new 'theorems' published every year. Perhaps we could take the collective works of mathematicians over the last 50 (500?) years and analyse them in some way?

Finally, there is Aquamarine's 'utility' question (the economics of mathematics? not to be confused with 'usefulness' - utility is a term in economics with a specific and precise meaning): the allocation of scarce resources (salaries for people to do math). I suspect that this is somewhat beyond the farthest shores of economics today ... for example, innovation has only recently begun to be decently addressed (economically, so to speak), and I suspect 'multi-generational' returns are a grey area. This matters not one bit at the individual level (there will always be those who are independently wealthy - free to pursue whatever interests them, and free to fund whomever they like, for example) - but it may be of great interest in the broad.

[selfAdjoint]

So, to what I see as the core of Aquamarine's post: of the truly vast potential 'space' mathematicians could explore (workers in the 'truth mines' - anyone read the SF novel "Diaspora"?), the regions most heavily explored have tended to be those with an apparent connection to 'models of the physical universe'. This is NOT to say that all (or nearly all) mathematicians work in areas that are seen to be (potentially) 'useful', nor that there aren't wonderful results far from the beaten track (e.g. Cantor's work?).

Perhaps we could use the scientific method to examine this idea? I read somewhere that there are ~250,000 new 'theorems' published every year. Perhaps we could take the collective works of mathematicians over the last 50 (500?) years and analyse them in some way?

A cheap and easy substitute for this would just to peruse the mathematical section of the arxiv. I'll bet a lot that you wouldn't find a majority of the papers there motivated by describing physical reality.

[jcsd]

It is an emprirical process, you learn about maths from the empirical observation of textbooks.

[CrankFan]

That question has been asked and answered earlier. And I did not claim that QM has falsified predicate logic.

When you wrote:

"There are some logical and mathematical systems that are more true than others"

What examples did you have in mind?

[Aquamarine]

A cheap and easy substitute for this would just to peruse the mathematical section of the arxiv. I'll bet a lot that you wouldn't find a majority of the papers there motivated by describing physical reality.
Maybe. But much mathematical research is not there. And probably almost all of that is applied. Much is never reveled. Like cryptography. The NSA is the world's largest employer of mathematicians with PhDs and has budget larger than the CIA. Or mathematics in military research. Or mathematics considered trade secrets by corporations. In finance mathematics is a central part of forecasting prices, often using exotic techniques.

And very importantly, everything involving programming computers is mathematical research into boolean algebra. All the steps from designing the CPU to high-level programming is an enormously complex application of boolean algebra. Including the research into efficient algorithms for database searches, data compression or graphical dispaly of 3D objects.

[Aquamarine]

When you wrote:

"There are some logical and mathematical systems that are more true than others"

What examples did you have in mind?
I would consider the non-Euclidean geometry behind GR more true than Euclidean geometry. Or predicate logic more true than the logic of Aristotoles. Based on that they have enabled theories with better accuracy in the real world. This does not mean that Euclidean geometry is not consistent using its own axioms. But that it is a less true theory regarding the real world.

[Hurkyl]

It might interest you to know that Euclidean geometry plays an essential role in the very definition of differential geometry (the geometry used by GR).

[CrankFan]

It is an emprirical process, you learn about maths from the empirical observation of textbooks.

Learning from texts is one way to learn about mathematics.

Is it impossible to develop a theorem, in your head, independent of any of the 5 senses?

I guess a lot of this depends on what is meant by "empirical" and "empirical process". If we are using a definition of empirical in which every thought process is an empirical process, then what do we gain by referring to a certain process (carried out by humans) as an empirical process?

[Aquamarine]

It might interest you to know that Euclidean geometry plays an essential role in the very definition of differential geometry (the geometry used by GR).
I am not a physicist or mathematician but I quess you mean that a manifold is like Euclidean space when near a point. But this does not mean that the Euclidean manifold gives better predictions than the pseudo-Riemannian manifold. Which would you say is a more true theory of the real world?

[Hurkyl]

Yes. In fact, the very definition of a manifold is that it is "locally Euclidean". Without Euclidean space, you do not have manifolds.


It is a somewhat unusual usage of the word "truth" to suggest that a theory can be "more true" than one of the components upon which it depends in an essential way.

And as another interesting bit of trivia, all of differential geometry can be done entirely within Euclidean space. In a very real sense, differential geometry is not a new theory, it's just a new way of looking at an old theory. This also puts your usage of the word "truth" in an awkward position.



It's clear that you intend "true" to refer, in some sense, as a sort of rating about how well a mathematical theory models the "real world", but I think you'll be very hard pressed to give it a precise meaning that is consistent with your usage. (let alone the "typical" usage of the term)

[jcsd]

What Aquamarine calls 'truer' systems, he means stronger systems in the sense of Goedel's second incomplteness theorum. Clearly that a system is stronnger than another syas nothing about their relative 'truth'.

[Aquamarine]

Yes. In fact, the very definition of a manifold is that it is "locally Euclidean". Without Euclidean space, you do not have manifolds.


It is a somewhat unusual usage of the word "truth" to suggest that a theory can be "more true" than one of the components upon which it depends in an essential way.

And as another interesting bit of trivia, all of differential geometry can be done entirely within Euclidean space. In a very real sense, differential geometry is not a new theory, it's just a new way of looking at an old theory. This also puts your usage of the word "truth" in an awkward position.



It's clear that you intend "true" to refer, in some sense, as a sort of rating about how well a mathematical theory models the "real world", but I think you'll be very hard pressed to give it a precise meaning that is consistent with your usage. (let alone the "typical" usage of the term)
Are you saying that it is not possible to tell if Euclidean or pseudo-Riemannian mathematical model gives better predictions in the real world? That you cannot tell which model is more true of the real world?

[Hurkyl]

Are you saying that it is not possible to tell if Euclidean or pseudo-Riemannian mathematical model

I am saying that anything that can be modelled by differentiable manifolds can be modelled equally well in Euclidean space.

This fact comes about because every n-dimensional differentiable manifold is homeomorphic to an n-dimensional surface in some higher dimensional Euclidean space.

Informally speaking, this means any statement about a differentiable manifold is true if and only if it is also true about the corresponding surface in Euclidean space.

Furthermore, if the manifold is Riemann, the surface can be chosen so the Riemann metric coincides with the Euclidean metric. For pseudo-Riemann manifolds, I think there's a corresponding theorem for Minowski space (which is just Euclidean space with a different dot product)

[Aquamarine]

I am saying that anything that can be modelled by differentiable manifolds can be modelled equally well in Euclidean space.

This fact comes about because every n-dimensional differentiable manifold is homeomorphic to an n-dimensional surface in some higher dimensional Euclidean space.

Informally speaking, this means any statement about a differentiable manifold is true if and only if it is also true about the corresponding surface in Euclidean space.

Furthermore, if the manifold is Riemann, the surface can be chosen so the Riemann metric coincides with the Euclidean metric. For pseudo-Riemann manifolds, I think there's a corresponding theorem for Minowski space (which is just Euclidean space with a different dot product)
You did not answer my questions.

[Hurkyl]

anything that can be modelled by differentiable manifolds can be modelled equally well in Euclidean space.

This doesn't?

[Aquamarine]

So you are saying that the both models are equally good for the real world?

[Nereid]

IF the 'real world' is one in which GR is 'true', then yes. The same holds for a many areas in physics ... the maths used is partly for convenience; the systems are 'formally equivalent'.

[Aquamarine]

Just to be certain. The real world is both Euclidean and non-Euclidean at the same time and both models are equally useful when making predictions about the real world.

[Hurkyl]

Yes.

A lower dimensional example of the embedding theorem might help for understanding.


The surface of a sphere is a non-Euclidean geometry, right? However, the sphere is embedded in Euclidean 3-space. Thus, any statement about the surface of the sphere is simultaneously a statement about Euclidean 3-space.


Conversely, Euclidean space is a differentiable manifold.



Addendum: IIRC, for the topologies that physicists consider physically reasonable for the universe, you don't even need to appeal to a higher dimensional space!

[Nereid]

I haven't the faintest idea what 'the real world' is! And I would argue that 'the real world' cannot be known through science (for some good discussions on this, and whether 'the real world' can be known through something other than science, see several active threads elsewhere in PF Philosophy).

The strongest statement one can make about successful physics (and science in general) theories is something like this: "within its domain of applicability, all results from good experiments and observations are consistent with {the theory}". Sometimes the domain of applicability is very large (e.g. the universe, in the case of GR); sometimes the testing is fairly shallow (e.g. only the 'weak field' and 'static' parts of GR have been tested); sometimes the degrees of accuracy of the tests are astonishing (e.g. QED has been tested to 12 (16?) decimal places); sometimes there is a 'huge' gap in what can be tested (e.g. no physical theories have been directly tested on distance scales smaller than ~10^-18m, and even indirectly only very weakly).

But there is no claim - in the theories themselves - that they are the real world.

Just in case ... the very good accuracy of (most) successful scientific theories means that you can post things on PF, buy useful things cheaply in your local shop that were made in factories on the other side of the world, have the choice to live to a 'ripe old age' (probabilistically), etc.

[Aquamarine]

I haven't the faintest idea what 'the real world' is! And I would argue that 'the real world' cannot be known through science (for some good discussions on this, and whether 'the real world' can be known through something other than science, see several active threads elsewhere in PF Philosophy).

The strongest statement one can make about successful physics (and science in general) theories is something like this: "within its domain of applicability, all results from good experiments and observations are consistent with {the theory}". Sometimes the domain of applicability is very large (e.g. the universe, in the case of GR); sometimes the testing is fairly shallow (e.g. only the 'weak field' and 'static' parts of GR have been tested); sometimes the degrees of accuracy of the tests are astonishing (e.g. QED has been tested to 12 (16?) decimal places); sometimes there is a 'huge' gap in what can be tested (e.g. no physical theories have been directly tested on distance scales smaller than ~10^-18m, and even indirectly only very weakly).

But there is no claim - in the theories themselves - that they are the real world.

Just in case ... the very good accuracy of (most) successful scientific theories means that you can post things on PF, buy useful things cheaply in your local shop that were made in factories on the other side of the world, have the choice to live to a 'ripe old age' (probabilistically), etc.
Would you agree that a theory that is more consistent with experiments is more true?

[Hurkyl]

I would not agree.

My primary objection is the vagueness of the term "true". I always strive to say what I really mean than invoke vague and often contentious terms.

Of course, if you defined "truth" as a measure of consistency with experiment, then I would certainly agree that a theory more consistent with experiment is more true, by your definition.


My next objection is that it does not make sense to ask if a mathematical theory is consistent with experiment. It does, however, make sense to ask if the interpretation of physical phenomena within a mathematical theory agrees with experiment.

[Aquamarine]

I would not agree.

My primary objection is the vagueness of the term "true". I always strive to say what I really mean than invoke vague and often contentious terms.

Of course, if you defined "truth" as a measure of consistency with experiment, then I would certainly agree that a theory more consistent with experiment is more true, by your definition.


My next objection is that it does not make sense to ask if a mathematical theory is consistent with experiment. It does, however, make sense to ask if the interpretation of physical phenomena within a mathematical theory agrees with experiment.
I can agree with much of what you say. Truth has been difficult to define in philosophy. It seems that it is not possible to decide semantic truth within in a formal language, but only using an outside language. That would seem to lead to the old infinite regress arguement.
http://en.wikipedia.org/wiki/Semantic_theory_of_truth

But the title of my thread was "Is mathematics empirical?". And I have seen nothing to contradict that. In the sense that what is studied is mainly decided by what can be of use in the real world, in the sense that it helps make better predictions about the real world. Those mathematics that are of no help in making predicions and will not help in making predictions in the future are frankly not any different from playing an interesting but useless game.

[matt grime]

You are misquoting my post.


You asked if some theories were more true than others without defining "true" and it's comparative usage here. Since most of us think something is true or it isn't, how is asking for some claification about this, and to get the debate away from seemingly pointless ideas, misquoting? Quoting out of context perhaps....


As Hurkyl points out, you're using true in some vague way that really isn't clear to us.

Every differential manifold of dimension N can be embedded isometrically in euclidean 2N-1 space if I recall my manifolds correctly, though I wouldn't put money on that.

Note, that lots of mathematical physicists don't even use manifolds, they use algebraic varieities.

[Aquamarine]

You asked if some theories were more true than others without defining "true" and it's comparative usage here. Since most of us think something is true or it isn't, how is asking for some claification about this, and to get the debate away from seemingly pointless ideas, misquoting? Quoting out of context perhaps....


As Hurkyl points out, you're using true in some vague way that really isn't clear to us.

Every differential manifold of dimension N can be embedded isometrically in euclidean 2N-1 space if I recall my manifolds correctly, though I wouldn't put money on that.

Note, that lots of mathematical physicists don't even use manifolds, they use algebraic varieities.
Let us abandon truth since there is no consensus of what it is and it seems not possible to define semantic truth within a formal language. Or at least discuss that in a different thread. Let us go back to the question that you edited out:

So why then is mathematics/logic interesting and why are some systems studied instead of others?

[Hurkyl]

But the title of my thread was "Is mathematics empirical?". And I have seen nothing to contradict that. In the sense that what is studied is mainly decided by what can be of use in the real world, in the sense that it helps make better predictions about the real world. Those mathematics that are of no help in making predicions and will not help in making predictions in the future are frankly not any different from playing an interesting but useless game.

Well, the sense in which you mean "empirical" differs from the sense in which I understand "empirical". One definition listed by Google (the others are similar) is:

Derived from experience or experiment.

There's a substantial difference between being motivated by experimental concerns, and actually derived from experiment.

[Aquamarine]

Well, the sense in which you mean "empirical" differs from the sense in which I understand "empirical". One definition listed by Google (the others are similar) is:



There's a substantial difference between being motivated by experimental concerns, and actually derived from experiment.
I have repeatedly said that the real world cannot say anything about the consistency of mathematical proofs, once the axioms have been chosen. My point is that most of those who do mathematical research are using axioms chosen because of usability in the real world.

[selfAdjoint]

My point is that most of those who do mathematical research are using axioms chosen because of usability in the real world.

And I continue to ask you to validate that statement by showing it to be true for an accessible data set of mathematical papers, e.g. the math section of the arxiv over thepast year. BTW, you won't often find a set of axioms in a math paper, what you find is references to previous papers.

[matt grime]

I would like to echo selfAdjoint there. The foundations of maths, whatever they may be, are of very little interest to mathematicians anymore. We do what we do. The vast majority of mathematics (including mathematical physics) papers have no reference to the world or experiment. In some sense that's what makes it hard to follow but it is also a necessity in order to make any progress at all. Are the underlying rules, rules that we make little reference to, chosen for their applicability to the real world? To be honest who knows, or cares - it's all a matter of interpretation anyway. For instance, do the axioms of ZF encapsulate the real world of "sets"? Many will say no. Indeed as soon as we start to do maths we create far more objects (used in a reasonably accurate sense for the category theorist) than exist, so the rules we need to manipulate them will not have any reflection in the real world: the set of all sets is purely a theoretical issue. There are many ways to pass beyond finite collections of things, and none of them can really be said to encapsulate the real world since there are only a finite number of objects in it, so how can we say the (very necessary) rules we choose there reflect anything 'real'?

If we take the axiom of choice as false then there are vector spaces without well defined bases. If we accept it then there are unmeasurable sets, and there is the Banach Tarski paradox to deal with.

Here is another discussion about such things:

http://www.maa.org/devlin/devlin%5F6%5F01.html

[Aquamarine]

I will repeat my previous post:

Maybe. But much mathematical research is not there. And probably almost all of that is applied. Much is never reveled. Like cryptography. The NSA is the world's largest employer of mathematicians with PhDs and has budget larger than the CIA. Or mathematics in military research. Or mathematics considered trade secrets by corporations. In finance mathematics is a central part of forecasting prices, often using exotic techniques.

And very importantly, everything involving programming computers is mathematical research into boolean algebra. All the steps from designing the CPU to high-level programming is an enormously complex application of boolean algebra. Including the research into efficient algorithms for database searches, data compression or graphical dispaly of 3D objects.

[Aquamarine]

I would like to echo selfAdjoint there. The foundations of maths, whatever they may be, are of very little interest to mathematicians anymore. We do what we do. The vast majority of mathematics (including mathematical physics) papers have no reference to the world or experiment. In some sense that's what makes it hard to follow but it is also a necessity in order to make any progress at all. Are the underlying rules, rules that we make little reference to, chosen for their applicability to the real world? To be honest who knows, or cares - it's all a matter of interpretation anyway. For instance, do the axioms of ZF encapsulate the real world of "sets"? Many will say no. Indeed as soon as we start to do maths we create far more objects (used in a reasonably accurate sense for the category theorist) than exist, so the rules we need to manipulate them will not have any reflection in the real world: the set of all sets is purely a theoretical issue. There are many ways to pass beyond finite collections of things, and none of them can really be said to encapsulate the real world since there are only a finite number of objects in it, so how can we say the (very necessary) rules we choose there reflect anything 'real'?

If we take the axiom of choice as false then there are vector spaces without well defined bases. If we accept it then there are unmeasurable sets, and there is the Banach Tarski paradox to deal with.

Here is another discussion about such things:

http://www.maa.org/devlin/devlin%5F6%5F01.html
And why then should society support those mathematics, if it is of no usefulness in the real world? Why not spend the money on teaching dead languages? Or on those prefering to play Everquest all day? Or on better health care? Or on reducing poverty? Why should taxpayer's money support those who want to play an useless game?

[master_coda]

And why then should society support those mathematics, if it is of no usefulness in the real world? Why not spend the money on teaching dead languages? Or on those prefering to play Everquest all day? Or on better health care? Or on reducing poverty? Why should taxpayer's money support those who want to play an useless game?

Money is spent on mathematics because there are people who are interested in its results. Just like any other type of research.

[Aquamarine]

Money is spent on mathematics because there are people who are interested in its results. Just like any other type of research.
Yes, but why should the state do that, using taxpayer's money? For those mathematics of no use in the real world? I am not arguing against indviduals spending their own money.

[HallsofIvy]

"The state" appears to believe that knowledge is a good thing in and of itself. It is also true,as I pointed out before, that often applications for pure mathematics are found AFTER the pure mathematics is done for no other reason than to "see how it works"- Riemannian Geometry was defined and researched long Einstein realized it could be used to describe general relativity.

By the way, the "state" doesn't support a heck of a lot of pure mathematics research. That typically is done by university professors who are paid to teach. The research is pretty much "on the side" (though an important side: it makes the university look better!). Wiles, for example, wasn't given any government support for his research into Fermat's Last Theorem (no doubt part of his salary was through government funding but it wasn't for the purpose of proving Fermat's Last Theorem)- which, by the way, is a good example of "non-empirical" mathematics.

Your original statement, by the way, was "truth in mathematics is ultimately derived from physics." Have you given up on that?

[CrankFan]

And why then should society support those mathematics, if it is of no usefulness in the real world?

Great idea!

Take non-Euclidean geometry. A perfect example of mathematics which had no obvious use in the "real world" at the time of its development. It's madness that anyone wasted their time with such fantasies when their time could have been equally well spent playing EQ!

And digital cryptography. More garbage. It just seems like many of the useful theorems of cryptography, discovered many centuries ago or earlier, have obvious application. The enlightened know that this is just an illusion; any result that follows from investigations into a branch of mathematics with no obvious application is necessarily useless in the "real world".

Just think about how much better off science would be if mathematicians never wasted any time on pure mathematics.

Oh wait a second...

[Aquamarine]

"The state" appears to believe that knowledge is a good thing in and of itself. It is also true,as I pointed out before, that often applications for pure mathematics are found AFTER the pure mathematics is done for no other reason than to "see how it works"- Riemannian Geometry was defined and researched long Einstein realized it could be used to describe general relativity.

By the way, the "state" doesn't support a heck of a lot of pure mathematics research. That typically is done by university professors who are paid to teach. The research is pretty much "on the side" (though an important side: it makes the university look better!). Wiles, for example, wasn't given any government support for his research into Fermat's Last Theorem (no doubt part of his salary was through government funding but it wasn't for the purpose of proving Fermat's Last Theorem)- which, by the way, is a good example of "non-empirical" mathematics.

Your original statement, by the way, was "truth in mathematics is ultimately derived from physics." Have you given up on that?
Yes, I have given upp a consensus opinion of truth, especially since it is not possible to define semantic truth within a formal language. But the title and essence still stands. Is mathematics empirical?

You do seem to agree that the value of apparently useless mathematics above playing a computer game is that they may be useful in the future. And my point is that this is what decides mathematical research, other people are not interested in supporting useless games played of mathematicians. In return for working for the mathematicians, they want something in return. So most mathematical research is forced into that useful in and agreeing with the real world. In that sense it is empirical. And do not argue for beauty or being like poetry, almost all people except of the mathematicians themselves do not understand what they are doing or producing.

If the state do not support much pure mathematics, then that is support for my view.

[master_coda]

Yes, but why should the state do that, using taxpayer's money? For those mathematics of no use in the real world? I am not arguing against indviduals spending their own money.

Because not very many people want the state to only spend money on things "of use in the real world".

[Aquamarine]

Because not very many people want the state to only spend money on things "of use in the real world".
You think so? That they do not want the state to concentrate its limited resources on things useful, like health care or reducing poverty? And it they do not want that, why should mathematics be supported instead of playing Everquest 24/7/52.

[Aquamarine]

Great idea!

Take non-Euclidean geometry. A perfect example of mathematics which had no obvious use in the "real world" at the time of its development. It's madness that anyone wasted their time with such fantasies when their time could have been equally well spent playing EQ!

And digital cryptography. More garbage. It just seems like many of the useful theorems of cryptography, discovered many centuries ago or earlier, have obvious application. The enlightened know that this is just an illusion; any result that follows from investigations into a branch of mathematics with no obvious application is necessarily useless in the "real world".

Just think about how much better off science would be if mathematicians never wasted any time on pure mathematics.

Oh wait a second...
That there were very little research into non-Euclidean geometry for centuries is support for my point. Not useful, not much research. Of course there will always be some who do research into apparently useless areas, sometimes with funds of their own. But most will be into areas apparently useful.

Regarding non-Euclidean geometry, research would have begun into that area anyway when it become clear that experiments were difficult to fit into Euclidean geometry. It is not evidence for that apparently useless mathematics must be done.

When it becomes clear that the old mathematical theories have difficulty fitting the facts of the real world, then new research will be done into now useful areas.

[master_coda]

You think so? That they do not want the state to concentrate its limited resources on things useful, like health care or reducing poverty? And it they do not want that, why should mathematics be supported instead of playing Everquest 24/7/52.

Almost everyone has their own little "useless" pet projects that they want to funnel public money into. Since there are lots of people who appreciate the acquisition of knowledge for its own sake, it's not surprising that a few crumbs of public money finds its way into the hands of mathematicians. Even if other people consider those endeavours useless.

And even then, it's usually indirect subsidies. People who are paid to do teach or do more applied research use some of their time doing pure math research. A scholarship to a student who's specializing in pure math. That sort of thing.


And of course, usefullness is in the eye of the beholder anyway. After all, health care money is sometimes spent keeping people alive who will no longer be able to contribute to society. So that money would be just as uselessly spent; but other people consider such work to be an end unto itself and would thus say that it is useful. Without a common standard for what is useless and what is not, usefullness isn't a good standard for public funding.

[master_coda]

When it becomes clear that the old mathematical theories have difficulty fitting the facts of the real world, then new research will be done into now useful areas.

This is like telling people to only invest in businesses that will be successful.

We don't know what areas of math are going to be helpful to other fields until after we've developed the math. How are we supposed to recognize an application of a math theory if we don't know anything about the theory yet because it's "useless"? We can't study the theory until we have an application, and we won't recognize the application until we study the theory.

And what's the threshold of difficulty before it becomes acceptable to research new fields? There isn't a lot of middle ground; requiring a concrete application in advance is basically equivalent to no research ever, and only requiring the possibility of an application makes almost any reasearch acceptable.

[Aquamarine]

Almost everyone has their own little "useless" pet projects that they want to funnel public money into. Since there are lots of people who appreciate the acquisition of knowledge for its own sake, it's not surprising that a few crumbs of public money finds its way into the hands of mathematicians. Even if other people consider those endeavours useless.

And even then, it's usually indirect subsidies. People who are paid to do teach or do more applied research use some of their time doing pure math research. A scholarship to a student who's specializing in pure math. That sort of thing.


And of course, usefullness is in the eye of the beholder anyway. After all, health care money is sometimes spent keeping people alive who will no longer be able to contribute to society. So that money would be just as uselessly spent; but other people consider such work to be an end unto itself and would thus say that it is useful. Without a common standard for what is useless and what is not, usefullness isn't a good standard for public funding.
Regarding a common standard, in practice most people are rule utilitarians. So spending money to reduce illness fit the common standard.

But you are right that people have somewhat different goals. And that therefore some money will find its way into not apparently useless mathematics. But that it is little money is support for the view that most people want to have something useful in return for working for the mathmaticians.

[Aquamarine]

This is like telling people to only invest in businesses that will be successful.

We don't know what areas of math are going to be helpful to other fields until after we've developed the math. How are we supposed to recognize an application of a math theory if we don't know anything about the theory yet because it's "useless"? We can't study the theory until we have an application, and we won't recognize the application until we study the theory.

And what's the threshold of difficulty before it becomes acceptable to research new fields? There isn't a lot of middle ground; requiring a concrete application in advance is basically equivalent to no research ever, and only requiring the possibility of an application makes almost any reasearch acceptable.
That is true for all investments. It is not possible to know before which one is the best and should get the resources. But it is possible to make educated guesses. So resources are given to those mathematicians who are working with axioms more probable to give useful results.

The need for new research can be when facts from experiments fit poorly with the known mathematical theories. Or doing research close to areas that given useful results before.

Again, this is like all other investments, most of those are done in areas mostly known before. Those investing in extremely speculative ideas, like a working black hole power generator, is likely to lose their investment. So little investing is done there.

[master_coda]

That is true for all investments. It is not possible to know before which one is the best and should get the resources. But it is possible to make educated guesses. So resources are given to those mathematicians who are working with axioms more probable to give useful results.


This is already the way things are done. The most heavily subsidized fields in mathematics are in fields with immediate applications e.g. cryptography. Some money is invested into general research without a specific purpose; this is hardly something unique to mathematics.


In fact, your argument doesn't seem to have anything to do with math at all. It seems to really just be a dispute about the proper way to invest public money.

[Aquamarine]

This is already the way things are done. The most heavily subsidized fields in mathematics are in fields with immediate applications e.g. cryptography. Some money is invested into general research without a specific purpose; this is hardly something unique to mathematics.


In fact, your argument doesn't seem to have anything to do with math at all. It seems to really just be a dispute about the proper way to invest public money.
Just a pointing out which areas and axioms of mathematics are most researched. And the reason why and that it is a good reason. Apparently, this is very controversial.
:cool:

[Aquamarine]

See also my earlier discussion that predicate logic and the current concepts of limits and derivatives could be abandoned in the future, if this would be useful. :tongue2:

[master_coda]

Just a pointing out which areas and axioms of mathematics are most researched. And the reason why and that it is a good reason. Apparently, this is very controversial.
:cool:

Well, the issue most people have is that you seem to think that something is invested in has to do with its validity or truthfulness, or how empirical a subject is. And refering to axioms is misleading; nobody invests in axioms.

Business investment strategies are hardly the place to be looking for truth. They're barely capable of producing adequate solutions to problems, even in the short-term.

[matt grime]

Regarding non-Euclidean geometry, research would have begun into that area anyway when it become clear that experiments were difficult to fit into Euclidean geometry. It is not evidence for that apparently useless mathematics must be done.

I've seen this view expressed a lot, without anyone ever providing any evidence to support it. How do you know that these physicists wojuld have thought to develop this theory? If it weren't there already why do you even think they would have gone towards it? It is just something that we use as a model after all, it isn't actually anything real.



I would like to proffer the alternate suggestion of why send people to the moon? Why develop satellite TV, why make particle accelerators? They are of no direct use. In fact practically none of the work in any physics department in the world today has any use in the real world. We are very good at predicting the behaviour of subatomic particles, and where there should be black holes, but physics can do squat about famine, drought, war, earth quakes, avalanches, aids....

It can create an atomic bomb, chemical weapons and food stuffs that cause chronic obesity, though. So, yes, we mathematicians are the ones that should hand our head in shame at how little we contribute to the world.


Returning to mathematics and leaving the devil's advocacy alone, many of us have removed the notion of limit from derivative and found it very useful in mathematics and physics already.

[HallsofIvy]

Please do not "hi-jack" other peoples threads to tout your own pet non-sense. You have posted this same thing on two of you own threads- I don't mind that so much but don't try to take over other peoplkes threads.

[selfAdjoint]

Shorter Ryskamp:

Can't do relativity in economics, so much for relativity.

I like Sraffa better than Keynes, so much for Keynes.

We in the 21st century are ever so much sharper than the big names of the 20th century, so much for them.

I have a thought on Einstein's relativity which is so confused it shows I don't understand Einstein. So much for me!

I am going to keep posting this same megillah until I cover the face of the earth.

[Philocrat]

There is no absolute truth in mathematics and logics. There are many consistent mathematical systems, consistent in the sense that they are not breaking the basic rules. But there is no absolute way of knowing that the basic rules are true. And there are limitless basic rules that can be chosen. Different basic rules gives different systems, like the intuitionist logic,predicate logic or fuzzy logic.

So why then is mathematics/logic interesting and why are some systems studied instead of others? I would argue it is because mathematics is empirical. There are some logical and mathematical systems that are more true than others. Those systems that more closely follow the real world are more true than others.

So truth in mathematics is ultimately derived from physics. Those mathematics that gives physicists more accurate models are more true.

Yes, it is!

[CrankFan]

Yes, it is!

Right, so how do you derive from physics, the truth that \pi is transcendental?

[honestrosewater]

Well, just to strengthen my point, do I understand correctly that the universe of discourse, x, is empty just in case [\exists x (Px)] is false and [\forall x (Px)] is true? BTW, I am not talking about the physical universe.

[CrankFan]

Well, just to strengthen my point, do I understand correctly that the universe of discourse, x, is empty just in case [\exists x (Px)] is false and [\forall x (Px)] is true? BTW, I am not talking about the physical universe.

In FOPL, the domain of discourse is non-empty, by definition. Have no idea what your point is.

[honestrosewater]

In FOPL, the domain of discourse is non-empty, by definition. Have no idea what your point is.My question about the universe of discourse is the subject of another thread (http://www.physicsforums.com/showthread.php?t=63775) where comments about the physical universe and the applications of logic keep intruding. Now that the question is here, I'm hoping those irrelevant comments will follow it.

[CrankFan]

My question about the universe of discourse is the subject of another thread (http://www.physicsforums.com/showthread.php?t=63775) where comments about the physical universe and the applications of logic keep intruding. Now that the question is here, I'm hoping those irrelevant comments will follow it.

I see your initial post in that thread, and questions along the lines of: if the domain of discourse is empty, is this or that formula "true" (I think you meant, a theorem).

But before anyone can answer your question they need to know what kind of logic you're talking about. If you're talking about first order predicate logic (which seems to me to be a reasonable assumption) then the answer to your original question is: that case never arises. So there's no point in taking it any further.

[honestrosewater]

CrankFan, so as not to intrude here, my reply is here (http://www.physicsforums.com/showpost.php?p=468161&postcount=18).

[Philocrat]

Right, so how do you derive from physics, the truth that \pi is transcendental?

A Superior Logic or Mathematics is the one that CAN reconcile SEQUENTIALISM with SIMULTANEITY. The mariage between these two beasts is the very foundation or seat of reality. My prediction is that there is no logical or quantitative utterances that can transcend this spooky marriage. The fate of Logic or mathematics lies in the continual interplay of these beasts in the spooky marriage. If reality colapses so will the marriage along with its underlying sustaining quantitative and logical components! That mathematics or logic is empirically enliven is beyond dispute.

[Hurkyl]

All I have to say is :confused: